We first examine the case of the time delay \(\tau _0=0\) , i.e., the two pulses act on the molecule simultaneously. The first amplitude condition of Eq. (REF ) is used by considering \(j=0\) , i.e., \(|\theta _1(t_f)|=0.3412\pi \)  and \(|\theta _2(t_f)|=0.3401\pi \) . The phases are fixed at \(\phi _1=\phi _2=-\pi /2\)  by Eq. (REF ). Figure REF  shows the dependence of the maximal orientation on the bandwidth \(\Delta \omega \)  and the detuning \(\Delta =\omega _{0}-\omega _{01}\) , in which the exactly calculated results by Eq. (REF ) are compared with the analytical results by using the first-order Magnus approximation. Both simulations show the maximum value of the orientation and its dependence on the parameters, and the exact result and the analytical match better and better as the bandwidth becomes narrower. To further show this change, Figs. REF  (b) and (e) shows the maximum degree of orientation versus the bandwidth by fixing \(\Delta =0\) , and Figs. REF  (c) and (f) exhibits its dependence on detuning by fixing the bandwidth at \(\Delta \omega =0.02/\tau ^{\prime }\)  with \(\tau ^{\prime }=\pi /2\omega _{01}\) .
In the broad bandwidth regime, the maximum orientation value is smaller than the theoretical maximum \(0.7746\) , and there are some slight differences between the exact and analytical results. The former can be attributed to the overlap between the frequency distributions of the two pulses. That is, the fundamental-frequency pulse with a broad bandwidths also results in transitions between the states \(|10\big \rangle \)  and \(|20\big \rangle \) , and the second harmonic one for the same reason also induces transitions between the states \(|00\big \rangle \)  and \(|10\big \rangle \) . The latter is caused by optical transitions beyond the rotational ladder climbing, as discussed in our previous work [1]}. The simultaneous two-photon transitions from \(|00\big \rangle \)  to \(|20\big \rangle \)  play a role.
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